Linearly Ordered Radon-nikodým Compact Spaces

We prove that every fragmentable linearly ordered compact space is almost totally disconnected. This combined with a result of Arvanitakis yields that every linearly ordered quasi Radon-Nikodým compact space is Radon-Nikodým, providing a new partial answer to the problem of continuous images of Radon-Nikodým compacta. It is an open problem posed by Namioka [8] whether the class of Radon-Nikodým compact spaces is closed under continuous images. Several authors [5] [2] [1, p. 104] who have studied this problem have introduced some superclasses of the class of Radon-Nikodým compacta which are closed under continuous images, although all these classes turned out to be equal to the class of quasi Radon-Nikodým compacta as shown in [9] and [3]. Let us recall that (1) A compact space K is Radon-Nikodým compact if and only if there exists a lower semicontinuous metric d : K ×K −→ [0,+∞) which fragments K. (2) A compact space K is quasi Radon-Nikodým compact if and only if there exists a lower semicontinuous quasi metric d : K × K −→ [0,+∞) which fragments K. (3) A compact space K is a fragmentable compact if and only if there exists a quasi metric d : K ×K −→ [0,+∞) which fragments K. Here, a quasi metric is a symmetric map d : K × K −→ [0,+∞) such that d(x, y) = 0 if and only if x = y but which may fail triangle inequality. Also, a map d : K × K −→ [0,+∞) is said to fragment the topological space K if for every nonempty (closed) subset L of K and every ε > 0 there exists a relative open subset U of L of diameter less than ε, that is, sup{d(x, y) : x, y ∈ U} < ε. Lower semicontinuity means that the set {(x, y) : d(x, y) ≤ a} is closed for every a ≥ 0. The class of fragmentable compacta is larger than the other two, for instance any Gul’ko non Eberlein compact is an example of fragmentable and not quasi Radon-Nikodým compact. It is again an open problem whether every quasi RadonNikodým compact is Radon-Nidkodým compact (as mentioned earlier, the class of quasi Radon-Nikodým compacta is closed under continuous images, and it is even 2000 Mathematics Subject Classification. 06A05, 54D30, 46B50, 46B26, 54D05.